1Cademy - Simplifying $$(2x^{-6}y^8)(-5x^5y^{-3})$$ and $$(3u^{-5}v^7)(-4u^4v^{-2})$$ Using the Product Property

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Negative Exponent

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Simplifying $(2x^{-6}y^8)(-5x^5y^{-3})$ and $(3u^{-5}v^7)(-4u^4v^{-2})$ Using the Product Property

Simplify products that involve both numerical coefficients and multiple variables with negative exponents by grouping similar components.

ⓐ $(2x^{-6}y^8)(-5x^5y^{-3})$ : Rewrite the expression by grouping the numerical coefficients and the like variable bases together: $2(-5) \cdot (x^{-6}x^5) \cdot (y^8y^{-3})$ . Multiply the coefficients ( $2(-5) = -10$ ) and add the exponents for each variable base: for $x$ , $-6 + 5 = -1$ ; for $y$ , $8 + (-3) = 5$ . This yields $-10x^{-1}y^5$ . Finally, use the negative exponent definition to move only the factor with the negative exponent ( $x^{-1}$ ) to the denominator: $\frac{-10y^5}{x}$ .

ⓑ $(3u^{-5}v^7)(-4u^4v^{-2})$ : Group similar components: $3(-4) \cdot (u^{-5}u^4) \cdot (v^7v^{-2})$ . Multiply the coefficients to get $-12$ . Add the exponents for $u$ ( $-5 + 4 = -1$ ) and for $v$ ( $7 + (-2) = 5$ ), resulting in $-12u^{-1}v^5$ . Apply the negative exponent definition to rewrite $u^{-1}$ in the denominator, giving $\frac{-12v^5}{u}$ .

Numerical coefficients are multiplied normally. Only variables that end up with negative exponents are relocated to the denominator.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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